Indeed, if (REF ) is false, then by the vanishing lemma [1]}, we know that \(\Vert {w}_{a}(x)\Vert _{r}\rightarrow 0\)  as \(a\nearrow a_s^{*}\)  for any \(2<r < 2_s^{*}\) , which contradicts (REF ).
Set
\(\bar{w}_a(x):=w_a(x+z_a),\) 
it then follows from (REF )-(REF ) by taking \(\bar{z}_{a}=\epsilon _{a}z_{a}\)  that (REF ) and (REF ) hold.
